Preparing approximate $N$-fold cat states with the phase space instruction set

Abstract

The phase space instruction set is a continuous-variable universal gate set involving single-qubit rotations and qubit-dependent displacements on a single boson. Using these gates, we prove that a circuit depth $\mathrmΩ(φ(N))$ is necessary to approximately prepare a large $N$-fold rotationally invariant Schrödinger cat state; here $φ(N) \gtrsim N/\log\log N$ is the Euler totient function. A protocol saturating this asymptotic bound on circuit depth is obtained for every prime number $N$. This protocol has an asymptotically optimal runtime, when the gates are generated by Hamiltonian evolution. Our results provide a sharp example where a universal gate set is surprisingly inefficient at preparing a simple family of states, and further imply that converting bosonic circuits between different universal gate sets can be extremely inefficient.